I wrote last autumn a chain of postings inspired by my noble attempts to understand the articles of Nima and collaborators. I certainly failed at the level of technical details but I think that I managed to abstract the essentials from TGD point of view. By combining twistorial ideas with the basic visions of TGD- in particular zero energy ontology and bosonic emergence- I ended up with very nice general picture. I give a list of my postings in chronological order below. The titles might give some idea how the ideas about the connection of twistors and TGD evolved (I had of course pondered the problem already earlier).
- What could be the generalization of Yangian symmetry?
- Exact Yangian symmetry, non-trivial scattering amplitudes, no IR singularities: only a dream?
- M4×CP2 from pairs of twistors and twistorial formulation of TGD
- A comment about exact Yangian symmetry realized in terms of bound states of partons
- A comment about formulation of TGD in the product of twistor space and its dual
- Yangian symmetry, twistors, and TGD
- Unitarity in TGD variant of Grassmannian twistor approach
This activity led to a new chapter Yangian Symmetry, Twistors, and TGD of "Towards M-Matrix" giving a more organized view about the final outcome.
For some time ago Lubos told about the latest articles from Nima and collaborators: A Note on Polytopes for Scattering Amplitudes and Local Integrals for Planar Scattering Amplitudes. Unfortunately, I did not have time to try to read the articles so that my frustrated reaction was just If I only had time.
Soon after this Lubos gave a link to a video in which Witten talked about knot invariants. This talk was very inspiring and led to TGD based vision about how to calculate invariants of braids, braid cobordisms, and 2-knots in TGD framework and the idea that TGD could be seen as symplectic QFT for calculating these invariances among other things. Much of work was just translation of the basic ideas involved to TGD framework.
One crucial observation was that one can assign to the symplectic group of δM4+ × CP2 gerbe gauge potentials generalizing ordinary gauge potentials in terms of which one can define infinite number of classical 2-fluxes allowing to generalize Wilson loop to a Wilson surface. Most importantly, a unique identification for the decomposition of space-time surface to string world sheets identified as singularities of induce gauge fields and partonic 2-surfaces emerged and one can see the two decompositions as dual descriptions. TGD as almost topological QFT concretized to a symplectic QFT for knots, braids, braid cobordisms, and 2-knots. These ideas are document in the following two postings and also in the brand new chapter Knots and TGD of "TGD: Physics as Infinite-Dimensional Geometry".
- Witten's talk about knot invariants
- Witten's physical view about Khovanov homology translated to TGD framework
I did not realize the obvious connection with twistor approach as I wrote the articles and new chapter.
In his rather energetic lecture Nima emphasized how the Yangian symmetry originally discovered in 2-D QFTs, algebraic geometry, twistor theory, and string theory fuse to something bigger called "T". I realized that the twistorial picture developed in the earlier postings integrates nicely with the braidy vision inspired by Witten's talk and that one could understand in TGD framework why twistor description, Yangian symmetry of 2-D integrable systems, and algebraic geometry picture are so closely related. In particular, the dual conformal symmetries of twistor approach could be understood in terms of duality between partonic 2-surfaces and string world sheets expressing the strong form of holography. Also a generalization for the dual descriptions provided by super Wilson loop and ordinary scattering amplitude in N=4 SUSY in terms of Wilson sheets suggests itself among many other things. Also a rather obvious solution to the problem posed by non-planar diagrams to twistor approach suggests itself. Planar diagrams are simply not present and parton-string duality and huge symmetries of TGD give good reasons for why this should be the case.
1. The declaration of revolution by Nima from TGD point of view
At first look Nima's program is a declaration of revolution against all that has been sacred to us (do not count me: I started my personal revolution about 33 years ago;-). Nima dooms space-time, Nima wants to get rid of QFT, Nima does not even care about unitarity, Nima wants to throw Feynman diagrams to paper basket (as the Europe's worst Feynman-graphic designer I think that much hotter place would be in order for a mathematical recipe which has produced so much suffering). Nima does not even respect string theory and sees it only as one particular- possibly inept- manner to describe the underlying simplicity.
- In many respects I agree with Nima about the fate of space-time of QFT. I however see Nima's view a little bit exaggerated: one can perhaps compute scattering amplitudes without Minkowski space but one cannot translate the results of computations to the language of experiments without bringing in frequencies and wavelengths, classical fields, and therefore also space-time. Quantum classical correspondence: this is needed and this brings space-time unavoidably into the picture. Space-time surface serves as a dynamical correlate for quantum dynamics- generalized Bohr orbit required by General Coordinate Invariance and strong form of holography. The enormously important implication is absence of Feynman graphs in ordinary sense since their is no path integral over space-time surface but just single surface: the preferred extremal of Kähler action is enough (forgetting the delicacies caused by the failure of classical determinism in standard sense for Kähler action allowing to realize also the space-time correlates of quantum jump sequences).
- Nima uses black hole based arguments to demonstrate that local observables are not operationally defined in neither gravitational theories nor quantum field theories and concludes that space-time is doomed. What would remain would be 4-D space-time regarded as a boundary of higher dimensional space-time (AdS/CFT correspondence). I think that this is quite too complex and that the reduction in degrees of freedom is much more radical: the landscape misery is after all basically due to the exponential inflation in the number of degrees of freedom due to the fatal mistake of making 10-D target space dynamical.
What remains in TGD are boundaries of space-time surfaces at the upper and lower ends of causal diamonds CD×CP2 (briefly CD) and wormhole throats at which the signature of induced metric changes from Euclidian to Minkowskian (recall that Euclidian regions represent generalized Feynman diagrams). CD is essentially a representation of Penrose diagram which fits nicely with twistor approach. Strong form of holography implies that partonic 2-surfaces (or dual string world sheets) and 4-D tangent space data a them are enough as basic particle physics objects. The rest of space-time is needed to realized quantum classical correspondence essential for quantum measurement theory.
The basic message of TGD is that quantum superpositions of space-time surfaces are relevant for physics in all scales. Particles are the dynamical space-time quanta. There is however higher-dimensional space-time which is fixed and rigid M4×CP2 and is needed for the symmetries of the theory and guarantees the Kähler geometric existence of the world of classical worlds (WCW). This simplifies the situation enormously: instead of 10- or 11-D dynamical space-time one has just 4-D space-time and 2-D surfaces plus 4-D tangent space data. Holography is what we experience it to be: we see only 2-D surfaces. And physics is experimental science although some super string theorists might argue something else!
- Nima argues also that fields are doomed too. I must say that I do not like this Planck length mysticism: it assumes quite too much and in TGD framework something new emerge already in CP2 scale about 104 longer than Planck scale. According to Nima all this pain with Feynman diagrams would be due to the need to realize unitary representations of Poincare group in terms of fields. For massless particles one is forced to assume gauge invariance to eliminate the unphysical polarizations. Nima sees gauge invariance as the source of all troubles. Here I do not completely agree with Nima. The unitary time evolution in fixed space-time translated to the path integral over classical fields is what leads to the combinatorial nightmare of summing over Feynman diagrams and plagues also φ4 theory. Amusingly, as Nima emphasizes all this has been known for 60 years. It is easy to understand that the possibility to realize unitarity elegantly using Feynman diagrams led to the acceptance of this approach as the only possible one.
In TGD framework the geometry of sub-manifolds replaces fields: the dynamics of partonic 2-surfaces identified as throats of light-like wormhole contacts containing fermions at them gives rise to bosons as bound states of fermions and antifermions. There is no path integral over space-time surfaces, just functional integral over partonic 2-surfaces so that path integral disappears (one cannot however exclude the possibility that the reduction of functional integrand to Chern-Simons action could allow the imaginary unit transforming functional integral to path integral). In zero energy ontology this means that incoming states are bound states of massless fermions and antifermions at wormhole throats and virtual states consist also of massless fermions but without the bound state constraint. This means horribly strong kinematic constraints on vertices defined by partonic 2-surfaces and UV finiteness and IR finiteness are automatic outcome of the theory. Massivation guaranteeing IR finiteness is consistent with massless-ness of fundamental particles since massive states are bound states of massless particles.
- Nima talks also about emergence as something fundamental and claims that also space-time emerges. In TGD framework emergence has very concrete meaning. All particles are bound states of massless fermions and the additional purely bosonic degrees of freedom correspond to vibrational degrees of freedom for partonic 2-surfaces.
2. Basic results of twistor approach from TGD point of view
The basic ideas of twistor approach are remarkably consistent with the basic picture of TGD.
2.1 Only on mass-shell amplitudes appear in the recursion formula
What is striking that the recursion formula of Nima and collaborators for the integrands of the planar amplitudes of N=4 SUSY involve only on mass shell massless particles in the role of intermediate states. This is in sharp conflict with not only Feynman diagrammatic intuition but also with the very path integral ideology motivated by the need to realize unitary time development.
As already mentioned, in ZEO (zero energy ontology) all states- both on mass shell and off mass shell are composites of massless states assigned to 2-D partonic surfaces. Path integral is indeed replaced with generalized Bohr orbits and one obtains only very few generalized Feynman diagrams. What remains is functional integral over 3-surfaces, or even less over partonic 2-surfaces with varying tangent space data. It was already noticed that reduction to Chern-Simons action allows also to consider path integral option.
A further simplification is that as a result of the dynamics of preferred extremals many particle states correspond to discrete sets of points at partonic 2-surfaces serving as the ends of orbits of braid strands and possibly also 2-knots and functional integral involves integral over different configurations of these points. The physical interpretation is as a realization of finite measurement resolution as a property of dynamics itself. The string word sheets are uniquely identified as inverse images under imbedding map of space-time surface to H=M4× CP2 of homologically non-trivial geodesic sphere of CP2 defining homological magnetic monopole. Holography in its strongest sense states that all information about non-trivial 2-homology if space-time surface and knottedness of the string world sheets is coded to the data at partonic 2-surfaces. For details see the chapter Knots and TGD of "TGD: Physics as Infinite-Dimensional Geometry".
2.2 Twistors and algebraic geometry connection emerge naturally in TGD framework
H=M4×CP2 and the reduction of all on mass shell states to bound states of massless states imply that twistor approach is the natural description of scattering amplitudes in TGD framework (see Yangian Symmetry, Twistors, and TGD of "Towards M-Matrix").
What is new that one must convolute massless theories in the sense that opposite throats of CP2 sized wormhole contacts carry massless states. This allows to get rid of IR divergencies and realize exact Yangian symmetry by a purely physical mechanism making particle states massive as described in Yangian Symmetry, Twistors, and TGD.
An important implication is that even photon, gluons, and graviton have small masses and that in TGD framework all components of Higgs field are eaten by electroweak gauge bosons. Also gluons have colored scalar and pseudo-scalar counterparts and already now there are some hints at LHC for pseudo-scalar gluons. The discovery of Higgs can of course kill this idea anytime.
The connection with twistors allows to understand how algebraic geometry of projective spaces emerges in TGD framework and one indeed ends up to an alternative formulation of quantum TGD with space-time surfaces in H replaced with holomorphic 6-surfaces of CP3×CP3 which are sphere bundles and there effectively 4-D. The equations determining the 6-surfaces are dictated by rather general constraints (see this).
2.3 Dual descriptions in terms of QFT and strings
The connections of N=4 SUSY with 2-D integrable systems and the possibly of both stringy and QFT descriptions characterized by dual conformal symmetries giving rise to Yangian invariance reduce in TGD framework to the duality between descriptions based on string world sheets and partonic 2-surfaces.
- The connection with string description emerges from the basic TGD in the sense that one can localize the solutions of the modified Dirac equation at braid strands located at the light-like 3-D wormhole throats. Similar localization to string world sheets defined in the above described manner holds true in space-time interior. The solutions of the modified Dirac equation localized to braid strands (and to string world sheets in space-time interior) are characterized by what I called pseudo momenta not directly identifiable as momenta. The natural identification is as the region momenta of the twistor approach. Recall that the twistorialization of region momenta leads to the momentum twistor approach making dual conformal invariance manifest.
- The strange looking localization of fermions at braid strands makes mathematically sense only because the classical dynamics of preferred extremals reduces to hydrodynamics such that the flow parameters for flow lines integrate to global coordinates. So called Beltrami flows are in question and mean that preferred extremals have interpretation as perfect fluid flows for which dissipation is minimal. This property implies also the almost topological QFT property of TGD meaning that Kähler action reduces to Chern-Simons action localized at light-like wormhole throats and space-like 3-surfaces at the ends of CDs.
- The mathematical motivation on braid strands comes from the fact that this allows to avoid delta functions in the anticommutators of fermionic oscillator operators at partonic 2-surfaces and therefore also the basic quadratic divergences of quantum field theories. The number of oscillator operators is finite or at least countable and the loss of complete locality is in terms of finite measurement resolution. The larger the number of braid points selected at partonic 2-surface, the larger the number string world sheets and the higher the complexity of space-time surface. This obviously means a concrete realization of holography. The oscillator algebra has interpretation as SUSY algebra with arbitrarily large N fixed by the number of braid points. This SUSY symmetry is dynamical and badly broken. For right handed neutrino the breaking is smallest but also in this case the mixing of left- and right handed M4 chiralities in modified Dirac equation implies non-conservation of R-parity as well as particle massivation and also the absence of lightest stable SUSY partner.
- The big difference between TGD and string models is that super generators do not correspond to Majorana spinors: this is indeed impossible for M4×CP2 since it would mean non-conservation of baryon and lepton numbers. I believed for a long time that stringy propagators emerge from TGD and the long standing painful question was what about stringy propagator defined by the inverse 1/G of the hermitian super generator in string models. In TGD 1/G cannot define stringy propagator since G carries fermion number. The reduction of strings to pairs of massless particles saves the situation and ordinary massless propagator for the counterparts of region momenta gives well defined propagators for on mass shell massless states! Stringy states reduce to bound states of massless particles in accordance with emergence philosophy. Nothing is scared these days!
2.4 Connection with integrable 2-D discrete systems
Twistor approach has revealed a striking connection between 2-D integrable systems and N=4 SUSY. For instance, one can calculate the anomalous dimensions of N=4 SUSY from an integrable model for spin chain in 2 dimensions without ever mentioning Feynman diagrams.
The description in terms of partonic 2-surfaces mean a direct connection with braids appearing in 2-D integrable thermodynamical systems and the description in terms of string world sheets means connection with integrable quantum field theories in 2-D Minkowski space. Both theories involve Yangian symmetry for which there exists a hierarchy of non-local conserved charged. Super-conformal invariance and its dual crucial for Yangian symmetry correspond to partonic 2-surfaces and string world sheets. The symmetry algebra is extended dramatically. In N=4 SUSY one has Yangian of conformal algebra of M4. In TGD this algebra is generalized to include the super Kac-Moody algebra associated with isometries of the imbedding space, the super-conformal variant of the symplectic algebra of δ M4×CP2, and also conformal transformations of M4 mapping given boundary of CD to itself.
This allows also to understand and generalize the duality stating that QFT amplitudes for N=4 SUSY have interpretation as supersymmetric Wilson loops in dual Minkowski space. The ends of braid strands indeed define Wilson loops. In TGD framework work one must however generalize Wilson loops to Wilson sheets and the circulations of gauge potentials are replaced with fluxes of gerbe gauge potentials associated with the symplectic group of δM4+× CP2. As noticed, dual conformal symmetries correspond to duality of partonic 2-surfaces and string world sheets implies by the 2-D holography for string world sheets.
3. Could planar diagrams be enough in the theory transcending N=4 SUSY?
Twistor approach as it appears in N=4 SYM is of course not the final solution.
- N=4 SUSY is not enough for the purposes of LHC.
- The extremely beautiful Yangian symmetry fails as one performs integration to obtain the scattering amplitudes and generates IR singularities. ZEO provides an elegant solution to this problem by replacing physical on mass shell particles with bound states of massless particles. Also string like objects emerge as this kind of states.
- Only planar diagrams allow to assign to assign to the sum of Feynman diagrams a single integrand defining the twistor diagram. Something definitely goes wrong unless one is able to treat the non-planar diagrams. The basic problem is that one cannot assign common loop momentum variables to all diagrams simultaneously and this is due to the tricky character of Feynman diagrams. It is difficult to integrate without integrand!
The easy-to-guess question is whether the sum over the non-planar diagrams vanishes or whether they are just absent in a theory transcending N=4 SUSY and QFTs. Let N denote the number of colors of the SUSY. For N → ∞ limit with g2N fixed only planar diagrams survive in this kind of theory and one obtains a string model like description as conjectured long time ago by 't Hooft. This argument led later to AdS/CFT duality.
The stringy diagrams in TGD framework could correspond to planar diagrams of N=4 QFT. Besides this one would have s functional integral over partonic 2-surfaces.
- The description would be either in terms of partonic 2-surfaces or string world sheets with both determined uniquely in terms of a slicing of space-time surface with physical states characterized in terms of string world sheets in finite measurement resolution.
- N → ∞ limit could in TGD framework be equivalent with two replacements. The color group with the infinite-D symplectic group of δM4+× CP2 and symplectic group and isometry group of H are replaced with their conformal variants.
- Could g2N =constant be equivalent with the use of hyper-finite factors of type II1 for which the trace of the unit matrix equals to 1 instead of N=∞. These factors characterize the spinor structure of WCW identifiable in terms of Clifford algebra defined by infinite-D fermionic oscillator algebra defined by second quantized fermions at partonic 2-surfaces.
4. Motives and twistors
Nima mentions at the end of his talk motives. I know about this abstract branch of algebraic geometry only that it is an attempt to build a universal cohomology theory , which in turn is an algebraic approach to topology allowing to linearize highly non-linear situations encountered typically in algebraic geometry where topology is replaced with holomorphy which is must more stringent property and allows richer structures.
- Physics as generalized number theory vision involving also the fusion of real and p-adic number fields to a larger super structure brings algebraic geometry to the core of TGD. The partonic 2-surfaces allowing interpretation as inhabitants of the intersection of real and p-adic worlds serve as correlates for living matter in TGD Universe. They are algebraic surfaces allowing in preferred coordinates a representation in terms of polynomials with rational coefficients. Motives would be needed to understand the cohomology of these surfaces. One encounters all kinds of problems such as counting the number of rational points in the intersection of p-adic and real variants of the surface and for algebraic surfaces this reduces to the counting of rational points for real 2-surface about which algebraic geometers know a lot of. Surfaces of form xn+ yn+ zn=0 for n≥ 3 appearing in Fermat's theorem are child's play since they allow only origin as a common point but the situation is not so easy in the general case;-).
- As cautiously concluded in Knots and TGD, the intersection form for string world sheets defines a representation of the second relative homology of space-time surface and by Poincare duality also second cohomology. "Relative" is with respect to ends of space-time (including restaurants) at the boundaries of CDs and light-like wormhole throats. The intersection form characterizing the collection of self-intersection points at which the badly treated braid strands are forced to go through each other is almost enough to characterize connected 4-manifolds topologically by Donaldson theorem.
- String world sheets define a violent unknotting procedure based on reconnections for braid strands- basic stringy vertex for closed strings- and in this manner knot invariant in the same manner as the recursion allowing to calculate the value of Jones polynomial for a given knot. Quantum TGD gives as a by-product rise to a symplectic QFT describing braids, their cobordisms, and 2-knots. It would not be surprising if the M-matrix elements would have also interpretation as symplectic covariants providing information about the topology of the space-time surface. The 2-braid theory associated with space-time surface would also characterize its topology just as ordinary knots can characterize topology of 3-manifolds.
To sum up, TGD suggests a surprisingly stringy but at the same time incredibly simple generalization of string model in which the discoveries made possible by the twistor approach to N=4 SUSY find a natural generalization. Nima has realized that much more than a mere discovery of computational recipes is involved and indeed talks about T-theory. Maybe he forgot two important letters "G" and "D" following the lonely "T" and completing it to holy trinity;-). I forgive this little lapsus linguae: Nima had a horrible hurry in trying to communicate as much as possible during single Nima's academic hour (100 minutes in standard time units;-)).